Download (680Kb) - Munich Personal RePEc Archive

MPRAMunich Personal RePEc Archive

Closed-Form Approximation of TimerOption Prices under General StochasticVolatility Models

Minqiang Li and Fabio Mercurio

Bloomberg LP, Bloomberg LP

2013

Online at http://mpra.ub.uni-muenchen.de/47465/MPRA Paper No. 47465, posted 10. June 2013 14:38 UTC

Closed-Form Approximation of Timer Option Prices under

General Stochastic Volatility Models

Minqiang Li∗, Fabio Mercurio†

April 2, 2013

Abstract

We develop an asymptotic expansion technique for pricing timer options under generalstochastic volatility models around small volatility of variance. Closed-form approximationformulas have been obtained for the Heston model and the 3/2-model. The approximation hasan easy-to-understand Black-Scholes-like form and many other attractive properties. Numer-ical analysis shows that the approximation formulas are very fast and accurate.

The contents of this article represent the authors’ views only and do not representthe opinions of any firm or institution.

∗Derivatives Research, Bloomberg LP. E-mail: [email protected].†Derivatives Research, Bloomberg LP. E-mail: [email protected].

1

1 Introduction

In 2007, Societe Generale Corporate and Investment Banking introduced a new variance deriva-

tive to the market called “timer options” (See Sawyer (2007)). A timer call option is similar

to a plain-vanilla call option, except that the expiry is not deterministic. Rather it is specified

as the first time when the accumulated realized variance exceeds a given budget. By allowing

a purchaser to custom choose the budget level, this product offers the purchaser a simple way

to combine directional bet and volatility bet within a single product. Major banks such as

Bank of America and UBS have been trading this product, sometimes under different names

such as “mileage option” or “mileage warrant”. Lehman Brothers at one point also published

a product overview of timer options (Hawkins and Krol, 2008). The buyers of timer options

are usually hedge funds, with the budget level often chosen below the accumulated variance

level that would occur with a constant implied volatility and a fixed target expiration in order

to lower the purchase price and to increase leverage. Sawyer (2008) also reported that Societe

Generale Corporate and Investment Banking started to sell other timer-style options such as

“timer out-performance options” and “timer swaps”.

Timer options were studied many years ago by Neuberger (1990) and Bick (1995) when such

security did not even exist in the market place. We note the recent enlightening theoretical

work of Carr and Lee (2010) and Lee (2012). The current paper deals with the computation

of timer option prices under a general stochastic volatility model. Despite the relative simple

payoff structure, pricing timer options turns out to be very challenging. One method to price

the timer option is through Monte Carlo simulation, as was studied in detail by Li (2010),

and Bernard and Cui (2011). Using the technique of time change, Bernard and Cui reduced

the computational cost of a single timer option from many hours to a few minutes, which is a

remarkable improvement. In addition to being time-consuming, there are other shortcomings

of the Monte Carlo method. For example, sensitivity analysis on model parameters usually

requires a complete re-run of the whole simulation. Also, often it requires large number of

simulations to get a relatively accurate estimate of the greeks. Another method by Liang,

Lemmens and Tempere (2011) involves multi-dimensional numerical integration of the Black-

Scholes formula where the transition density is given by a complex Fourier inversion which

involves special functions such as Bessel functions. This approach is very sophisticated in that

it uses the path integral technique developed in quantum field theory. However, the result

involves high-dimensional numerical integration, with one of the integration dimension involv-

ing complex Fourier inversion and special functions. The method gives exact result, but only

works for very limited number of models where the transition density is known. It is also com-

putationally expensive due to high-dimensional numerical integration and might have stability

2

issues similar to the complex integration in Heston model for plain-vanilla options. A third

method is to use techniques such as perturbation to get an accurate and fast approximation.

Recently, Saunders (2010) considers an asymptotic expansion for stochastic volatility models

under fast mean-reversion. The nice feature of this approximation is that it is in closed form.

However, a key shortcoming of this approximation is that it requires extremely large mean-

reversion coefficient to obtain satisfying accuracy. Table 1 of Saunders (2010) uses a value of

200 for this coefficient. In real life applications, it is usually in the order of 1 for equities or

indices and much smaller for some other asset classes. This limits its potential application.

In this paper, we take a different asymptotic expansion approach. Instead of looking at the

mean-reverting coefficient as in Saunders (2010), we consider an expansion on the volatility

coefficient of the instantaneous variance process. The reason is that when this coefficient is zero,

timer option has a closed-form solution. By proposing a simple Black-Scholes-like approximate

formula, we decompose the second-order pricing PDE of the timer option into three PDEs.

The first two PDEs are for the effective discounting maturities in the Black-Scholes formula

for the stock and strike terms. The third PDE is for the effective variance budget used in

the moneyness parameters in the Black-Scholes formula. It turns out that to low orders in

the volatility coefficient, all three PDEs can be solved analytically to give exact closed-form

formulas. Numerical analysis shows that our method is extremely fast and very accurate.

Implemented rather naively in MATLAB on Intel E8400, we are able to price a thousand

timer options within a couple of seconds. For the parameters used in Liang, Lemmens and

Tempere (2011), we get percentage pricing errors around 0.03% under the Heston model, and

around 0.20% under the 3/2 model across different strikes and different correlations.

Our method adds to the large literature of using asymptotic expansion in option pricing.

Such an expansion involves a small parameter, often taken to be a small quantity in the problem

such as volatility, time to maturity, mean-reverting strength or its reciprocal, etc. Oftentimes

searching for such a small quantity is crucial for the approximation.1 The current work is most

closely related to Lewis (2000), where he develops a small volatility of variance expansion for

plain-vanilla options under the Heston model. A big difference is that Lewis works under the

Fourier-transformed momentum space, while our analysis is done completely with the original

variables. We also note that small volatility expansion is also studied in Liption (2001). The

idea of using Black-Scholes-like formula to approximate option prices is also not new, a notable

example here being Kirk’s approximation formula for spread options (see Kirk (1995)).

Our approximation has many attractive properties. It resembles the Black-Scholes formula

1For example, Li, Deng and Zhou (2008) developed an expansion based on the curvature of the exercise conditionof two-asset spread option, and on the smallness of the Hessian matrix in the multi-dimensional spread option case.

3

which makes it easier to interpret. Because of this Black-Scholes-like form, greeks such as

delta and gamma are given in very simple forms. The approximation is fast and accurate.

It reduces to the known solvable cases, for example, when variance process is deterministic,

or when interest rate and dividend rate are both zero. It always gives positive option prices

and satisfies obvious arbitrage bounds. The approximation respects put-call parity, meaning

that the approximate prices for timer cash contract, time share contract, and timer puts are

consistent with the approximate timer call price. The method applies to generic stochastic

volatility models. As long as the drift and diffusion functions of the variance process are

simple, we can obtain a closed-form approximation. For all the papers on timer options cited

in this introduction, dividend rate is assumed to be zero for easier analysis. This is a strong

limitation in application. In this paper, we allow for nonzero dividend rate. An additional

attractive feature of our approximation is that the approximation formula does not break down

if the absolute value of the correlation coefficients of the two Brownian motions is around 1,

or even is exactly 1. This contracts the Monte Carlo method in Bernard and Cui (2011) where

a naive implementation with a perfect correlation would cause numerical problems. Finally,

as byproducts, our approach also gives an approximation for the expected exercise time of the

timer option and the expected stock price at the random exercise time. In fact, we are able to

get an approximation for the joint moment generating function of the random exercise time

and the stock price at the random exercise time.

Similar to the case of plain-vanilla options, we propose for the first time a concept of Black-

Scholes implied volatility for timer options. In contrast to the plain-vanilla options, for timer

options the implied volatility enters the Black-Scholes formula through the two discounting

factors rather than through the total variance. We then define the Black-Scholes implied

volatility surface for timer options. The availability of a fast closed-form formula allows us

to easily examine these surfaces for the Heston model and the 3/2 model. Such surfaces are

useful in practice when studying the effect of various parameters on timer option prices with

different variance budgets and different strike prices in a collective way.

The rest of the paper is organized as follows. Section 2 discusses the pricing of timer options

under the general stochastic volatility framework. We list three exactly solvable cases. We

then discuss the general perturbation technique based on the pricing PDE. We also specialize

the general technique to the Heston model and the 3/2 model. Section 3 makes several im-

portant remarks on our approximation. Section 4 contains numerical analysis of the proposed

approximations. It also looks at the Black-Scholes implied volatility surfaces for timer options

under the Heston model and the 3/2 model. Section 5 concludes. More detailed mathematical

derivations are in Appendix.

4

2 Pricing Timer Options

2.1 The Pricing Problem

We consider a general stochastic volatility framework for the stock process St and instantaneous

variance process Vt under the risk-neutral measure:

dSt = (r − δ)St dt+√

VtSt dWSt , (1)

dVt = a(Vt) dt+ ηb(Vt) dWVt . (2)

Here we assume that the interest rate r and dividend rate δ are both constants, and the two

Brownian motions W St and W V

t have a constant correlation coefficient ρ. The drift and diffu-

sion functions a(V ) and b(V ) are assumed to be functions of V only. This general framework

incorporates the well-known Heston model (Heston, 1993) as a special case, as well as the 3/2

model (see, for example, Ahn and Gao (1999)) . Notice we introduced a parameter η in the

diffusion function for V . We will call η the volatility of variance coefficient and assume that it

is small.

Now define the accumulated variance process to be

ξt = ξ +

∫ t

0Vu du. (3)

Here ξ0 = ξ is the accumulated variance at time 0. Notice that

dξt = Vtdt. (4)

A timer call option pays (Sτ − K)+ with a random remaining maturity τ . Here τ is the

random time remaining for a pre-specified variance budget B to be exceeded. Let the current

time be 0 with initial stock price S, initial instantaneous variance V , and initial realized total

variance ξ. If the contract is issued today, then ξ = 0, but we allow for the situation where the

timer call was issued some time in the past, so that ξ > 0. Except for when considering the

boundary conditions, we assume that ξ < B so that the timer call has not expired. Since ξt is

a continuous process, the random remaining maturity τ is given by

τ ≡ inf {t > 0 : ξt = B} = inf

{t > 0 :

∫ t

0Vu du = B − ξ

}. (5)

The processes St, Vt and ξt form a Markovian system which is sufficient to model the timer

option payoff. Therefore, we denote the price of the timer call today by C(S, ξ, V ). By risk-

neutral pricing, we have

C(S, ξ, V ) = E0

[e−rτ (Sτ −K)+

]. (6)

5

The timer put option pays (K − Sτ )+ at random maturity τ , and we have

P (S, ξ, V ) = E0

[e−rτ (K − Sτ )

+]. (7)

They satisfy the put-call parity

C(S, ξ, V )− P (S, ξ, V ) = E0

[e−rτSτ

]−KE0

[e−rτ

]. (8)

Here, the first term on the right-hand side is today’s price of a timer share contract, and the

second term is the price of a timer cash contract.

It is difficult to perform the expectation for timer call options because the formula involves

the expectation of a function of two dependent random variables τ and Sτ . Because of this

difficulty, we take a different route. Our starting point is the following pricing PDE for

C = C(S, ξ, V ) under the general stochastic volatility model:

V Cξ + a(V )CV + (r − δ)SCS +1

2η2b2(V )CV V +

1

2S2V CSS + ρηS

√V b(V )CSV − rC = 0.

(9)

The subscripts here denote partial derivatives. The boundary condition is

C(S,B, V ) = (S −K)+. (10)

The boundary conditions for S and V are similar to those for European style plain-vanilla

options and usually do not need to be specified explicitly. The above PDE is valid for any

European style derivative whose payoff is determined by the state variables S, ξ and V . In

particular, timer share contract, time cash contract, and time puts also satisfy this PDE.

2.2 Exactly Solvable Cases

Below we will mainly focus on the timer call. There are a few simple cases where the pricing

PDE can be solved exactly. These exact solutions provide clues on developing an approximation

technique. They are also useful for sanity checks once an approximation formula is obtained.

A good approximation should reduce to the exactly solvable cases when the conditions are

met.

2.2.1 Case 1: K = 0 and δ = 0

In this limiting case, the timer call option becomes a timer share contract with a random

maturity. Assuming that under the general stochastic volatility model the quantity e−rtSt is

a true martingale,2 the solution is then given by

C(S, ξ, V ) = E0

[e−rτSτ

]= S. (11)

2 Note this put restrictions on the drift and diffusion functions a(V ) and b(V ). See Section 1.2 of Zeliade (2011)and the references therein for the discussion of this issue for the Heston model.

6

It is easy to see that this solution satisfies the pricing PDE since C only depends linearly on S

and

rSCS − rC = 0. (12)

Notice also that this solution is model independent. That is, in this case the dynamics of the

instantaneous variance process does not affect the timer call option price. Notice that if δ is

not zero, then in general we do not have C(S, ξ, V ) = S.

2.2.2 Case 2: r = δ = 0

In this case, the exact solution is given by (see Bernard and Cui (2011), or Lee (2012))

C(S, ξ, V ) = SN(d+)−KN(d−), (13)

where

d± =log(S/K)√

B − ξ± 1

2

√B − ξ. (14)

Here N(·) is the cumulative normal distribution function. Notice that the solution does not

depend on V . It does not depend on ρ or η. It is easy to check that the above solution satisfies

the simplified PDE since

Cξ +1

2S2CSS = 0. (15)

Notice also that the solution for this r = δ = 0 case is model independent. That is, it does

not depend on the dynamics of the instantaneous variance.

2.2.3 Case 3: η = 0

When η = 0, the instantaneous variance process is deterministic, so we know exactly when

we are going to exercise the timer call. The solution C(S, ξ, V ) in this case reduces to the

Black-Scholes formula for plain-vanilla options:

C(S, ξ, V ) = Se−δTN(d+)−Ke−rTN(d−), (16)

with

d± =log(Se(r−δ)T /K)√

B − ξ± 1

2

√B − ξ. (17)

Here T = T (ξ, V ) is the solution of the first-order PDE

V Tξ + a(V )TV + 1 = 0, (18)

7

with the boundary condition

T (B,V ) = 0. (19)

Unlike the previous two solutions, this solution is model dependent as the solution of T depends

on a(V ).

It’s a little bit tedious but still quite simple to check that with T = T (ξ, V ), the above

solution satisfies the simplified PDE

V Cξ + a(V )CV + (r − δ)SCS +1

2S2V CSS − rC = 0. (20)

Instead of verifying that C(S, ξ, V ) with variance budget exceeding time T (ξ, V ) given above

satisfies the pricing PDE, we could also directly prove that T (ξ, V ) satisfying the above first-

order PDE is the variance budget exceeding time. See Appendix for details.

2.3 The Approximation Technique

2.3.1 A Particular Solution Form

We propose the following solution form for the timer call price in a general stochastic volatility

model:

C(S, ξ, V ) = Se−δT ′

N(d+)−Ke−rTN(d−), (21)

where

d± ≡ d±(S, T, T ′,Σ) =log(S/K) + rT − δT ′

Σ± 1

2Σ. (22)

Here we assume T = T (ξ, V ), T ′ = T ′(ξ, V ), and Σ = Σ(ξ, V ). That is, they have no

dependence on S. We will comment more on this assumption later.

This particular solution form is very attractive for many reasons. It is homogeneous of

degree 1 in S and K, as it should be. It is easy to verify that this solution form will reduce

to the three exactly solvable cases above if the total variance Σ2 reduces to B − ξ. Also, as

long as Σ is positive, we always have C(S, ξ, V ) > 0. This can be seen from the fact that

C(0+, ξ, V ) = 0 and CS > 0 for all S > 0. The formula has sensible limits when S or K goes

to 0 or +∞. If Σ → 0 when ξ → B, and Σ → +∞ if B → +∞, the formula will also have

sensible limiting behavior for B. Also, it is easy to check that we have kept the attractive

property in the Black-Scholes formula

Se−δT ′

n(d+)−Ke−rTn(d−) = 0. (23)

8

Here n(·) is the standard normal probability density function. This property is very attractive

because it simplifies the derivatives of C(S, ξ, V ), especially for the Greeks delta and gamma.

They are given by

∆ ≡ CS = e−δT ′

N(−d+), (24)

Γ ≡ CSS =1

SΣe−δT ′

n(d+). (25)

Finally, as we discuss later, the put-call parity for timer options has a simple form closely

resembling that of the plain-vanilla options:

C(S, ξ, V )− P (S, ξ, V ) = Se−δT ′ −Ke−rT . (26)

The proposed form is nonetheless more complicated than the small vol expansion for plain-

vanilla options under the Heston model as was done in Lewis (2000). In that case, T and T ′

are both constants, and the only approximation is through an expansion for Σ. In our case,

because the exercise time is random, T and T ′ will in general be different. They are differen

for two reasons. First, if we write E0eλτ = eλTeff , then in general Teff will depend on λ. The

second reason is that in the expectation E0Sτe−rτ , Sτ and e−rτ are dependent. Therefore, we

should expect that T does not depend ρ, but T ′ does. We could also have defined quantities

D(ξ, V ) = e−rT and D′(ξ, V ) = e−δT ′

and perform expansion on these two quantities. We

prefer the current approach of performing the expansion inside the exponents. This has the

slight benefit of guaranteeing the positivity of the two discount factors.

One possible alternative is to use the following expansion

C(S, ξ, V ) ≈ C0(S, ξ, V ) + ηC1(S, ξ, V ) + η2C2(S, ξ, V ) + · · · , (27)

where C0(S, ξ, V ) is the exact solution for η = 0 in equation (16). We do not take this route

for several reasons. First, In Lewis (2000), it is shown that this form of expansion for given

order of η is not as accurate as the expansion using Σ, especially for away-from-the-money

options. Expansion in Σ works better because the two cumulative normal functions act as

some sort of regularizer. Second, the above expansion can violate positivity in some cases,

and obscure the put-call parity. As we will see later, the expansion we choose allows a very

intuitive interpretation. For example, the term e−rT gives an approximation to second-order

in η for the price of a timer cash contract where cash is to be received at the random variance

budget exceeding time. Finally, our expansion form allows us to decompose the problem of a

complicated PDE for C(S, ξ, V ) into three PDEs for T , T ′ and Σ, which as we show later, are

often exactly solvable to low orders in η.

9

2.3.2 The PDEs

Our starting point is the pricing PDE for a timer call, which we rewrite here

V Cξ + a(V )CV + (r − δ)SCS +1

2η2b2(V )CV V +

1

2S2V CSS + ηρS

√V b(V )CSV − rC = 0.

(28)

The idea of our approximation is very simple. We take the derivatives of the trial solution

and put them into the PDE and try to satisfy the PDE as best as we can. This technique

of satisfying the PDE asymptotically was very fruitfully utilized in Aıt-Sahalia (1999), Aıt-

Sahalia (2002), and Aıt-Sahalia (2008) to derive transition density approximations for diffusion

processes. See also Li (2010) for the exact sense in which the backward and forward equations

of the transition density are satisfied.

By utilizing equation (23) and the fact that

d+ − d− = Σ, (29)

and

d±Σ = −d∓

Σ, d+Σ − d−Σ = 1, (30)

the partial derivatives we need can be readily computed as

Cξ =− δT ′ξSe

−δT ′

N(d+) + rTξKe−rTN(d−) + ΣξSe−δT ′

n(d+), (31)

CV =− δT ′V Se

−δT ′

N(d+) + rTV Ke−rTN(d−) + ΣV Se−δT ′

n(d+), (32)

CS = e−δT ′

N(d+), (33)

CSS =1

SΣe−δT ′

n(d+), (34)

CSV =− δT ′V e

−δT ′

N(d+) + d+V e−δT ′

n(d+), (35)

CV V = δ[δ(T ′

V )2 − T ′

V V

]Se−δT ′

N(d+)− r[r(TV )

2 − TV V

]Ke−rTN(d−)

+ CnV V Se

−δT ′

n(d+), (36)

where the derivatives of d± with respect to V are given by

d+V ≡ rTV − δT ′V − d−ΣV

Σ, (37)

d−V ≡ rTV − δT ′V − d+ΣV

Σ, (38)

and we define

CnV V ≡ −δT ′

V d+V + rTV d

−V +

[ΣV V − δT ′

V ΣV − ΣV d+d+V

]. (39)

10

In order for the PDE to be satisfied for all possible values of K, we postulate that the terms

proportional to N(d+), N(d−), and n(d+) should all be zero. Collecting the N(d+) terms and

simplifying gives a PDE for T ′(ξ, V ) as follows

V T ′ξ + a′(V )T ′

V +1

2η2b2(V )

[T ′V V − δ(T ′

V )2]+ 1 = 0, (40)

with the boundary condition

T ′(B,V ) = 0. (41)

Here for notational convenience we have also defined

a′(V ) ≡ a(V ) + ηρ√V b(V ). (42)

Collecting the N(d−) terms gives a PDE for T (ξ, V ) as follows

V Tξ + a(V )TV +1

2η2b2(V )

[TV V − r(TV )

2]+ 1 = 0, (43)

with the boundary condition

T (B,V ) = 0. (44)

The n(d+) terms are more complicated, and we have a PDE for Σ(ξ, V ) as follows

V Σξ + a(V )ΣV +V

2Σ+ ηρ

√V b(V )d+V +

1

2η2b2(V )Cn

V V = 0, (45)

with the boundary condition

Σ(B,V ) = 0. (46)

The boundary conditions on T , T ′ and Σ are chosen such that when ξ = B, the timer call

option price reduces to (S −K)+ for any K ≥ 0.

By looking at the PDEs, we notice that the assumptions that T and T ′ do not depend on S

are consistent with their PDEs. The assumption that Σ does not depend on S is not consistent

with its PDE as the PDE involves terms such as d+V which is a function of S. However, as we

will see later, for any stochastic volatility model, if we only solve Σ to first order in η, then

Σ will not be a function of S. This is what we do in this paper since we find that first-order

approximation in Σ is very satisfying in terms of accuracy. If one wants to solve Σ to higher

orders in η, equation (45) would have to be modified to include terms proportional to ΣS , ΣSS,

ΣSV , etc.

When r = 0, we do not need to consider the PDE for T and should just set rT = 0 in the

approximation formula. Similarly, when δ = 0, we do not need to consider the PDE for T ′.

11

Notice also that when δ = 0, the PDE for T does not involve ρ, so the effect of ρ is only

captured in the PDE for Σ. In particular, this means that the simple treatment of just taking

Σ2 = B − ξ would not be sufficient for nonzero ρ.

So far, no approximation has been made except that we postulate a particular form of

solution. The PDEs for T , T ′ and Σ are quite complicated in that they are all nonlinear and

interrelated. Below we describe our approach to solve them asymptotically to lowest orders

in η.

2.3.3 The Perturbation Approach

Motivated by the small volatility of variance expansion for stochastic volatility models in Lewis

(2000), we also seek an expansion in η for T , T ′ and Σ. The solution strategy of the above

interrelated PDEs is to solve them to zeroth order in η first, and then to perturb around these

zeroth-order solutions.

For T and T ′, we first aim to solve the following two first-order PDEs:

V T0,ξ + a(V )T0,V + 1 = 0, (47)

V T ′0,ξ + a′(V )T ′

0,V + 1 = 0. (48)

These two equations can often be solved exactly in time-homogeneous stochastic volatility

models. We then solve the following two first-order PDEs

V Tξ + a(V )TV +1

2η2b2(V )

[T0,V V − r(T0,V )

2]+ 1 = o(η2), (49)

V T ′ξ + a′(V )T ′

V +1

2η2b2(V )

[T ′0,V V − δ(T ′

0,V )2]+ 1 = o(η2). (50)

Notice that in the above equations, we put the error o(η2) on the right-hand side. It is easy to

see that the solutions of the above equations satisfy the original PDEs to second-order in η.

We want to make an important remark for the treatment of T ′ above. Because a′(V )

is linear in η, the solution T ′0 rigorously speaking is the approximate solution of T ′ to first

order in η. However, we will treat a′(V ) as raw model input and absorb the η dependence

inside a′(V ). Therefore, we use the notation T ′0 instead of T ′

1. In equation (50), the difference

of using the zeroth-order or first-order approximations in the η2 term will not affect the PDE

to order η2. The reason we choose to do this is that in many models, a(V ) and a′(V ) will

formally be identical. Therefore, the current treatment allows us to only solve one PDE instead

of two PDEs.

The solution for Σ when η = 0 is simply Σ =√B − ξ. Despite a simple zeroth-order

solution, the PDE for Σ is more complicated than those for T and T ′ when η is not zero. One

12

of the reasons is that the PDE for Σ involves derivatives of T and T ′. It turns out that in

many cases we can solve the PDE for Σ exactly to first order in η. Therefore, we aim to solve

V Σξ + a(V )ΣV +V

2Σ+ ηρ

√V b(V )

rTV − δT ′V − d−ΣV

Σ= O(η2). (51)

Approximating Σ to second order in η is much more difficult. First, we will be forced to

introduce S dependency in Σ. Second, the PDE for Σ to second order gets more complicated.

Fortunately, as we will see later, approximating Σ to first order in η gives very satisfying

accuracy in the models we consider.

The above PDE can be further simplified. Notice that TV − T0,V = O(η2). It can also be

shown that the solutions from the first two equations satisfy T ′V − TV = O(η). In addition,

we postulate that ΣV = O(η), which is fairly reasonable to expect since Σ0 is not a function

of V . These two statements are easy to verify in the two concrete models we consider later.

Collecting only the leading linear terms in η and multiplying the PDE above by 2Σ, the PDE

for Σ becomes a PDE for Σ2

V (Σ2)ξ + a(V )(Σ2)V + V + 2ηρ(r − δ)√V b(V )T0,V = O(η2). (52)

We could have used T ′0,V in place of T0,V in the equation above while still keeping O(η2)

accuracy, but because the term (Σ2)V involves a(V ), it is most convenient to use T0,V here.

The solution Σ20(ξ, V ) = B − ξ solves the PDE to zeroth order in η with the boundary

condition Σ20(B,V ) = 0. That is, we have

V (Σ20)ξ + a(V )(Σ2

0)V + V = 0. (53)

Notice that this zeroth-order solution is model-independent. Now assume a first-order expan-

sion of Σ2 in η given by

Σ2(ξ, V ) = B − ξ + 2ηρ(r − δ)G(ξ, V ). (54)

In order to satisfy equation (52) to first-order in η, we need G(ξ, V ) to solve the following PDE

V Gξ + a(V )GV +√V b(V )T0,V = 0, (55)

with the boundary condition

G(B,V ) = 0. (56)

Equations (49), (50) and (55) are the main results of this section. Once we obtain their

solutions, the final approximate timer option price is then given by the simple approximation

13

form in equation (21). These three equations look a little bit complicated on first sight, but

formally they can all be solved exactly. Notice that all three equations have the form of

V Qξ +A(V )QV = q(ξ, V ) (57)

for some functions A(V ) and q(ξ, V ) with the boundary condition Q(B,V ) = 0. This equation

can be solved in general by switching to the characteristic coordinates and changing the PDE

to an ODE. In Appendix, we explain this approach in more detail.

2.4 Specializing to the Heston Model

2.4.1 The PDEs

We now specialize the general approach in the last section to specific stochastic volatility

models. We first consider the well-known Heston model. In this model, a(V ) = κ(θ− V ), and

b(V ) =√V . So that we have

dS = (r − δ)Sdt+√V S dW S, (58)

dV = κ(θ − V )dt+ η√V dW V , (59)

where dW S · dW V = ρdt. We assume that the usual Feller condition 2κθ > η2 is satisfied

for the variance process, and that κ− ρη > 0. The Feller condition assures that the origin is

unattainable. The quantity κ−ρη comes out from performing a measure change on the Heston

model by using stock as the numeraire. See Section 1.4 of Zeliade (2011) and the references

therein for more discussions on this.

The PDEs for T and T ′ become

V Tξ + κ(θ − V )TV +1

2η2V

[TV V − r(TV )

2]+ 1 = 0, (60)

V T ′ξ + κ′(θ′ − V )T ′

V +1

2η2V

[T ′V V − δ(T ′

V )2]+ 1 = 0. (61)

Here κ′ = κ− ρη, and κ′θ′ = κθ. The change to κ′ and θ′ is due to the term ηρb(V )√V . This

adjusts for the fact that in Sτe−δτ , the two terms Sτ and e−δτ are dependent. As commented

in the last section, we absorb the η dependence in a′(V ) and treat κ′ and θ′ as raw model

inputs. Notice that these two equations are now formally identical except that in the equation

for T ′, we replace κ, θ and r with κ′, θ′ and δ. This allows us to only solve the PDE for T and

the solution for T ′ can be obtained by just replacing κ, θ and r with κ′, θ′ and δ.

The PDE for Σ becomes

V Σξ + κ(θ − V )ΣV +V

2Σ+ ηρV d+V +

1

2η2V Cn

V V = 0. (62)

14

2.4.2 Solving T and T ′

To zeroth-order in η, we need to solve

V T0,ξ + κ(θ − V )T0,V + 1 = 0, (63)

with the boundary condition

T0(B,V ) = 0. (64)

It turns out that T0(ξ, V ) is the implicit solution of

θT0 + (V − θ)1− e−κT0

κ= B − ξ. (65)

Appendix contains a proof of the above equation.

The implicit solution in equation (65) is not very convenient to use later on when looking

for higher-order solutions. It turns out that we can express T0 explicitly using Lambert’s

product log function W (x) which is defined as the unique solution of

W (x)eW (x) = x. (66)

The solution for T is then given by

T0(ξ, V ) =1

κlogR, (67)

where

R ≡ z0z

= ez−z0+κB−ξ

θ , (68)

z0 ≡V − θ

θ, (69)

z ≡ W(z0e

z0 · e−κB−ξ

θ

). (70)

Notice that we give two expressions for R. While the definition of z0/z is easier to understand,

it has a removable singularity at V = θ, where both z and z0 are zero. On the other hand, the

second expression is always well-defined.

Using the second expression for R, an equivalent way to write T which does not have the

apparent singularity at V = θ is

T0(ξ, V ) =z − z0

κ+

B − ξ

θ. (71)

This formula in preferred in the actual computation rather than equation (67).3 Appendix

contains a proof of equations (67) and (71).

3 The apparent singularity in equation (67) at V = θ is removable since near x = 0, we haveW (x) ≈ x−x2+O(x3).Indeed, in this case, we have the result T0 = (B − ξ)/θ from the alternative expression.

15

The zero-order solution for T ′ is similar. We have

T ′0(ξ, V ) =

1

κ′logR′ =

z′ − z′0κ′

+B − ξ

θ′, (72)

where

R′ ≡ z′0z′

= ez′−z′

0+κ′ B−ξ

θ′ , (73)

z′0 ≡V − θ′

θ′, (74)

z′ ≡ W(z′0e

z′0 · e−κ′ B−ξ

θ′

). (75)

To second order in η, we need to solve the following first-order PDEs:

V T ′ξ + κ′(θ′ − V )T ′

V +1

2η2V

[T ′0,V V − δ(T ′

0,V )2]+ 1 = 0, (76)

V Tξ + κ(θ − V )TV +1

2η2V

[T0,V V − r(T0,V )

2]+ 1 = 0. (77)

Let us focus on the equation for T first. Having obtained the solution for the η = 0 case,

we seek a solution of the form

T (ξ, V ) ≈ T0(ξ, V ) + η2H(ξ, V ;κ, θ, r) (78)

for some function H(ξ, V ). Plugging this solution into the PDE for T and collecting terms

proportional to η2, we need to solve a first-order PDE

V Hξ + κ(θ − V )HV +1

2V[T0,V V − r(T0,V )

2]= 0, (79)

with the boundary condition

H(B,V ) = 0. (80)

Although the above PDE involves the complicated terms T0,V V and (T0,V )2, it turns out

that this equation can be solved exactly to give

H(ξ, V ;κ, θ, r) =(R − 1)

[− r(1 + z)(1 + 2R2z +R(2z − 3)) + κ(2R2z2 +R(2− 5z − 2z2)− 2− z)

]

4κ3R2(1 + z)3θ

+

[3kz + r(2z2 + z − 1)

]logR

2κ3(1 + z)3θ. (81)

Notice that z and R are functions of ξ and V . A proof is given in Appendix.

Similarly, we have

T ′(ξ, V ) ≈ T ′0(ξ, V ) + η2H(ξ, V ;κ′, θ′, δ). (82)

Notice in H(ξ, V ;κ′, θ′, δ) we also need to replace z and R by z′ and R′ in addition to replacing

κ, θ and r with κ′, θ′ and δ.

16

2.4.3 Solving Σ

To first order in η, we need to solve the following first-order PDE

V (Σ2)ξ + κ(θ − V )(Σ2)V + V + 2ηρV (r − δ)T0,V = 0. (83)

with the boundary condition

Σ2(B,V ) = 0. (84)

In Appendix, we show that to first order in η, the approximation for Σ2 is given by

Σ2 = B − ξ +2ηρ(r − δ)

κ2· (1−R)(Rz − 1) +R(z − 1) logR

R(1 + z). (85)

The above formula is very interesting because it allows us to examine the effect of ρ qual-

itatively. In particular, it’s not necessarily true that a negative ρ will make the timer option

more expensive. Notice that |ρ| ≈ 1 does not impose any computational problem in our

approximation.

Compared with the Black-Scholes formula, the slight additional cost for timer call options

is to compute the product log z. Once z is computed, T , T ′ and Σ are simple functions of z

given in equations (78), (82), and (85).

2.5 Specializing to the 3/2 Model

2.6 The PDEs

In this model, the variance process is specified as an inverse Feller process as follows

dV = κV (θ − V ) dt+ ηV3

2 dW V . (86)

Therefore, a(V ) = κV (θ−V ) and b(V ) = V3

2 . Same as in the Heston model, dW S ·dW V = ρdt.

Using Ito’s lemma, it can be verified that the reciprocal of V follows a Cox-Ingersoll-Ross

process which is the case for the Heston model.

The PDEs we need to solve are

V T ′ξ + κ′V (θ′ − V )T ′

V +1

2η2V 3

[T ′V V − δ(T ′

V )2]+ 1 = 0, (87)

V Tξ + κV (θ − V )TV +1

2η2V 3

[TV V − r(TV )

2]+ 1 = 0, (88)

V Σξ + κV (θ − V )ΣV +V

2Σ+ ηρV 2d+V +

1

2η2V 3Cn

V V = 0. (89)

Here, we have

κ′ = κ− ρη, θ′ =κθ

κ′. (90)

17

Notice that the equation for T ′ has a similar modification as in the Heston model due to the

correlation between the two Brownian motions driving the stock process and the instantaneous

variance process.

2.6.1 Solving T and T ′

In this 3/2 model, T0 can be worked out explicitly to give

T0(ξ, V ) ≡ 1

κθlog

(V + θ(eκ(B−ξ) − 1)

V

). (91)

It is easy to check that T0(ξ, V ) satisfies the zeroth-order PDE

V T0,ξ + κV (θ − V )T0,V + 1 = 0. (92)

Similarly, the zeroth-order solution for T ′ is

T ′0(ξ, V ) ≡ 1

κ′θ′log

(V + θ′(eκ

′(B−ξ) − 1)

V

). (93)

To second order in η, we need to solve

V Tξ + κV (θ − V )TV +1

2η2V 3

[T0,V V − r(T0,V )

2]+ 1 = 0. (94)

In Appendix, we show that the solution is

T ≈ T0(ξ, V ) + η2H(ξ, V ;κ, θ, r), (95)

with

H(ξ, V ;κ, θ, r) =1− 4R + [3− 2 logR]R2

4κ3 [V + θ(R− 1)]2r

+4V[1 +

(logR− 1

)R]+ θ[− 3 + (4− 4 logR)R+ (2 logR− 1)R2

]

4κ2 [V + θ(R− 1)]2,

(96)

where to simplify the expression a little bit we have defined

R = eκ(B−ξ). (97)

It is easy to check that H(B,V ;κ, θ, r) = 0, so the approximate solution T (ξ, V ) satisfies the

boundary condition T (B,V ) = 0.

Similarly, the solution for T ′ is

T ′ ≈ T ′0(ξ, V ) + η2H(ξ, V ;κ′, θ′, δ). (98)

Notice in the above expression for H(ξ, V ;κ′, θ′, δ), we also need to use R′ = eκ′(B−ξ) instead

of R.

18

2.6.2 Solving Σ

The PDE we need to solve in the 3/2 model is

V (Σ2)ξ + κV (θ − V )(Σ2)V + V + 2ηρ(r − δ)V 2T0,V = O(η2). (99)

It turns out that this equation can be solved exactly to order η, yielding

Σ2 = (B − ξ)− 2ηρ(r − δ)

κ2· 1 + (logR− 1)R

V + θ(R− 1). (100)

It is easy to check that Σ satisfies the boundary condition Σ = 0 when ξ = B. The proof

of the above solution is given in Appendix. We note also that |ρ| ≈ 1 does not impose any

computational problem in the above approximation.

3 Remarks

3.1 Put-Call Parity

Before we turn to numerical analysis, let us make a few important observations. Readers who

are interested in numerical results can skip to the next section directly.

So far we only considered timer call options. The timer put option price can be worked

out similarly, and is given by the approximation

P (S, ξ, V ) ≈ Ke−rTN(−d−)− Se−δT ′

N(−d+) (101)

with exactly the same d− and d+ as those in the approximation for timer calls. Therefore, the

put and call prices satisfy the put-call parity in our approximation

C(S, ξ, V )− P (S, ξ, V ) = Se−δT ′ −Ke−rT . (102)

Here Se−δT ′

is the approximate time 0 price of receiving one share of stock at a random future

time τ , and Ke−rT is the approximate time 0 price of receiving cash K at time τ . By repeating

the analysis we have just done, it can be shown that both approximations Se−δT ′

and Ke−rT

satisfy the pricing PDE to order η2. In terms of risk-neutral expectations, we have

E0e−rτSτ = S0e

−δT ′

+ o(η2), (103)

E0e−rτ = e−rT + o(η2). (104)

19

3.2 Forward Price at Random Exercise Time τ

Some quantities of interest for timer options come out as byproducts of our approximation.

One quantity of interest is the forward price at the random exercise time. Denote it by

F = F (S, ξ, V ) = E0Sτ . (105)

Then it is easy to show that it satisfies the following PDE

V Fξ + a(V )FV + (r − δ)SFS +1

2η2b2(V )FV V +

1

2S2V FSS + ρη

√V b(V )SFSV = 0, (106)

with the boundary condition

F (S,B, V ) = S. (107)

Notice that equations (103) and (104) do not necessarily imply that F = SerT−δT ′

+ o(η2). If

we postulate an approximate form

F (S, ξ, V ) ≈ Se(r−δ)TF (ξ,V ), (108)

then similar algebra as before shows that in order to satisfy the PDE of F to second order

in η, we need TF to solve

V TFξ + a′(V )TF

V +1

2η2b2(V )

[TFV V − (r − δ)(TF

V )2]+ 1 = 0, (109)

with the boundary condition

TF (S,B, V ) = 0. (110)

This is the same approximate PDE for T ′ but with the dividend rate δ replaced by δ − r.

Therefore, in our approximation, we automatically get an approximation for F .

In both the Heston model and 3/2 model, we have that

F = E0Sτ = Se(r−δ)[T ′

0(ξ,V )+η2H(ξ,V ;κ′,θ′,δ−r)] + o(η2), (111)

where we should use the expressions of T ′0 and H for the respective models. For the Heston

model, they are given in equations (71) and (81). For the 3/2 model, they are given in

equations (91) and (96). Notice that the above equation is exact when η = 0.

3.3 Moments of Random Exercise Time τ

We first consider the expected time to exercise. A holder of the timer option may be interested

in this for risk management purposes. Denote this quantity by

TE(ξ, V ) ≡ E0τ. (112)

20

From diffusion process theory (see, for example, Karlin and Taylor (1981)), TE satisfies the

following PDE exactly

V TEξ + a(V )TE

V +1

2η2b2(V )TE

V V + 1 = 0, (113)

with the boundary condition

TE(B,V ) = 0. (114)

This is exactly the same PDE for T but with r = 0. Therefore, setting the solution for T with

r = 0 will also give us an approximation for TE to order η2. Specifically, for the Heston model,

we have

TE = E0τ = T0(ξ, V ) + η2H(ξ, V ;κ, θ, 0) + o(η2), (115)

where T0 is given in equation (71) and H is given in equation (81). The same equation is true

in the 3/2 model if we replace T0 and H with their counterparts in equations (91) and (96).

An easier way to derive the result above in these two models is to notice that equation (104)

actually gives us an approximation for the moment generating function Mτ (µ) because the

dynamics of Vt has no relation with the interest rate r. Notice that for any real number µ, the

function H(ξ, V ;κ, θ,−µ) can be decomposed into a sum of two terms under both models

H(ξ, V ;κ, θ,−µ) = H0(ξ, V ;κ, θ) + µH1(ξ, V ;κ, θ). (116)

Here H0 and H1 are not functions of µ. Equation (104) then says that

Mτ (µ) = E0eµτ = eµT

E+µ2η2H1(ξ,V ;κ,θ). (117)

This is precisely the moment generating function of a normal random variable. The mean can

be read off to be TE and the variance of τ is approximated as

Var(τ) = 2η2H1(ξ, V ;κ, θ) + o(η2). (118)

That is, for both the Heston model and the 3/2 model, our approximation effectively treats τ

as a normal random variable with mean and variance approximated to second order in η.

When ρ = 0, timer option price can be written as a mixing of Black-Scholes prices where

the mixing is an expectation on the random exercise time τ . See for example, Theorem 3.4 in

Bernard and Cui (2011). Assuming normality for τ , we can actually perform this expectation

in closed form to get an approximation formula which is only very slightly different from

the current one. Unreported numerical analysis shows that this approximation is also very

accurate. However, the current PDE approach has the advantage of being applicable for

nonzero ρ.

21

3.4 The Joint Moment Generating Function of logSτ and τ

3.4.1 The Perturbation Approach

For two real numbers λ and µ, the joint moment generating function of logSτ and τ is defined

as

MlogSτ ,τ (λ, µ) ≡ E0 eλ logSτ+µτ . (119)

Our perturbation method can be used to get an approximation for MlogSτ ,τ (λ, µ) to order η2.

This could be useful to value other exotic derivatives with a timer feature.

The moment generating function can be viewed as the expectation of a discounted payoff

of a power contract with discount rate −µ. Define this expectation as

Π(S, ξ, V ;λ, µ) = E0 Sλτ e

µτ . (120)

Then Mlog Sτ ,τ (λ, µ) = Π(S, ξ, V ;λ, µ). We will often just write Π(S, ξ, V ) when there is no

ambiguity. By Feynman-Kac theorem, Π satisfies the following PDE

V Πξ + a(V )ΠV + (r − δ)SΠS +1

2η2b2(V )ΠV V +

1

2S2VΠSS + ρηS

√V b(V )ΠSV + µΠ = 0,

(121)

with the boundary condition

Π(S,B, V ) = Sλ. (122)

Same as before, we postulate a particular solution form

Π(S, ξ, V ) = Sλep(ξ,V ). (123)

Notice here p(ξ, V ) has no S dependence. This actually can be justified on price homogeneity

ground. Plugging this solution into the PDE for Π, we get a nonlinear PDE for p(ξ, V ):

V pξ + aλ(V )pV + αV + β +1

2η2b2(V )

[pV V + (pV )

2]= 0, (124)

with the boundary condition

p(B,V ) = 0. (125)

Here we have defined

α ≡ 1

2λ(λ− 1), (126)

β ≡ (r − δ)λ+ µ, (127)

aλ(V ) ≡ a(V ) + λρη√V b(V ). (128)

22

Same as before, we will treat aλ(V ) as raw model input and absorb the η dependence. The

PDE for p(ξ, V ) can be solved exactly to second order in η. Write

p(ξ, V ) ≈ p0(ξ, V ) + η2J(ξ, V ). (129)

Then the zeroth-order solution (actually first-order in η since aλ(V ) is linear in η) p0 satisfies

the following first-order PDE

V p0,ξ + aλ(V )p0,V + αV + β = 0 (130)

with the boundary condition

p0(B,V ) = 0. (131)

After we solve for p0, the first-order PDE for J is given by

V Jξ + aλ(V )JV +1

2V[p0,V V + (p0,V )

2]= 0 (132)

with the boundary condition

J0(B,V ) = 0. (133)

The equations for p0 and J can be solved by using the method of characteristic coordinates,

as we did before. Notice that equations (103) and (104) are special cases of the above general

solution by setting λ = 0 or λ = 1.

Below we give the solutions for both the Heston model and the 3/2 model.

3.4.2 Solution for the Heston Model

The joint moment generating function above under the Heston model might not be defined for

all λ ∈ R. The condition for E0Sλt < ∞ for all t > 0 in the Heston model has been analyzed

in Andersen and Piterbarg (2007) which puts a restriction on ρ. To our knowledge, the well-

definedness of the joint moment generating function above has not been analyzed. Intuitively,

because τ is the random time the cumulated variance reaches B− ξ, it should require a weaker

condition to exist than in Andersen and Piterbarg (2007). In what follows, we assume that η

is small and κ− λρη > 0.

Define the following three quantitites

Rλ ≡ zλ0zλ

= ezλ−zλ0+κλ

B−ξθλ , (134)

zλ0 ≡V − θλ

θλ, (135)

zλ ≡ W

(zλ0e

zλ0 · e−κλB−ξ

θλ

), (136)

23

where the parameters κλ and θλ are defined by

κλ = κ− λρη, (137)

θλ = κλθλ/κ. (138)

Notice that we assume that κλ > 0. In this case, we also have θλ > 0. These two parameters

come out from aλ(V ) = κλ(θλ − V ).

The solution of p0(ξ, V ) is given by

p0(ξ, V ) =αθλ + β

θλ(B − ξ) +

β

κλ(zλ − zλ0). (139)

Notice that when λ = 0, α = 0 and β = 1, the solution reduces to that of T0. If λ = 1, α = 0

and β = 1, the solution reduces to that of T ′0. If α = 1 and β = 0, the solution reduces to Σ0.

The solution for J(ξ, V ) is given by

J(ξ, V ) = βH(ξ, V ;κλ, θλ,−β), (140)

with the function H exactly the same as in equation (81). This is easy to see since the PDE

for J is formally identical to the PDE for H because the first term in p0 has no V dependence.

Putting everything together, we have

MlogSτ ,τ (λ, µ) = Sλ exp(p0(ξ, V ) + βH(ξ, V ;κλ, θλ,−β)

)+ o(η2). (141)

The results for E0Sτ , E0τ , Var(τ), E0e−rτ , E0Sτe

−rτ and Mτ (µ) we have obtained earlier all

come out from the above result easily. For example, to evaluate E0Sτe−rτ , we just need to set

λ = 1 and β = (r − δ)− r = −δ in the above formula for MlogSτ ,τ (λ, µ).

We remark that besides these quantities, the above result on the joint moment generating

function also allows us to get approximations for useful quantities such as E0 log Sτ , Var(Sτ ),

Corr(Sτ , e−rτ ), etc.

3.4.3 Solution for the 3/2 Model

We will only give the result here without details. Define κλ and θλ same as in the Heston

model above. The solution of p0(ξ, V ) is given by

p0(ξ, V ) = α(B − ξ) +β

κλθλlog

(V + θλ(e

κλ(B−ξ) − 1)

V

). (142)

Same as in the Heston model, the solution for J(ξ, V ) is given by

J(ξ, V ) = βH(ξ, V ;κλ, θλ,−β), (143)

with the function H exactly the same as in equation (96).

24

4 Numerical Analysis

4.1 Accuracy Comparison

Before we look at the accuracy performance, we emphasize that the approximation we develop

in earlier sections is extremely fast. When implemented in MATLAB on a PC with an Intel

E8400 CPU, we are able to price a thousand timer options in a couple of seconds. It is possible

to have additional time saving when pricing a basket of timer options. For example, notice

that T , T ′ and G does not depend on K or ρ. Therefore, if we need to price otherwise similar

options with different K or different ρ, we do not need to recompute them.

Table 1 reports the accuracy comparison for the Heston model. Parameters used here are

the same ones as in Liang, Lemmens and Tempere (2011): V0 = 0.087, κ = 2, θ = 0.09,

η = 0.375, B = 0.087, ξ = 0, r = 0.015, S0 = 100, and δ = 0. This corresponds to a

target expected exercise time of roughly about 1 year. Analytical results are taken from

Liang, Lemmens and Tempere (2011), which are obtained through multi-dimensional numerical

integration. All errors reported are percentage errors with respect to the analytic results.

In Panel A, we give the timer option prices assuming that r = 0 and η = 0, where we

have exact pricing formulas. These exact solutions do not depend on ρ. As we see, even

with a relatively low interest rate of 1.5%, the naive approximation of assuming r = 0 gives

very inaccurate results, with the relative pricing errors ranging from 4.48% to 8.17%. On the

other hand, for this set of parameters, the η = 0 approximation performs much better than the

r = 0 approximation, especially for positive ρ. The relative errors exhibit strong ρ dependence,

indicating that the effect of ρ is important.

In Panel B, we compare the analytic results with the approximation in this paper. Approx1

uses prices obtained if we solve T , T ′ and Σ all to first order in η. In Approx1-2, Σ is solved

exactly to first order in η, while T and T ′ are solved exactly to second order in η. As we

see, Approx1 sufficiently captures the effect of ρ on timer options. The relative errors for

different ρ are now fairly close and are around 0.70%. The accuracy is significantly improved

if we use the second-order approximation for T and T ′. The relative errors for Approx1-2 are

now around 0.03%. This should be satisfying for most real-life applications.

Table 2 reports the accuracy comparison for the 3/2 model. Parameters used here are the

same ones as in Liang, Lemmens and Tempere (2011): V0 = 0.2952, κ = 22.84, θ = 0.46692,

η = 8.56, B = V0, ξ = 0, r = 0.015, S0 = 100, and δ = 0. Again, this roughly corresponds

to an expected exercise time of 1 year. Analytical results are taken from Liang, Lemmens

and Tempere (2011). Comparing Tables 1 and 2, we see that with the chosen parameters, the

analytic call option prices are very close for these two different models.

25

Panel A indicates that the r = 0 approximation is still very inaccurate. It also shows

that the η = 0 approximation is now less accurate than in the Heston model in Table 1. It

is difficult to have a fair comparison of η under two different models. One method is to look

at the long-run variance. If exists, a diffusion process with drift function µ(V ) and diffusion

function σ(V ) has a stationary density given by

π(V ) =M(v)

σ2(V )exp

(∫ V

v

2µ(u)

σ2(u)du

). (144)

Here v is an arbitrary lower integration limit, and M(v) is a constant depending on v so that

the density π(V ) integrates to 1. With the parameters in the two tables, the long-run means

under the Heston model and the 3/2 model are 0.084 and 0.090, respectively. The long-run

variances are given by 0.0032 and 0.0099, respectively. Therefore, the variance process is more

volatile in the 3/2 model than in the Heston model with the given parameters.

In Panel B, we compare the analytic results with the approximation in this paper. We see

that although Approx1 corrects for the effect of ρ in the right direction, it does not improve

the accuracy over the η = 0 approximation much. However, Approx1-2 improves the accuracy

significantly. Despite a much larger η, our approximation gives relative errors of around 0.2%.

The accurate analytic approximate formula allows us to quickly examine the effect of various

parameters on the timer call option price. Such a sensitivity analysis also provides a sanity

check for our approximation formula. Table 3 illustrates such an analysis for the Heston model.

The base case uses the same parameters as in Table 1. We also fix K = 110 and ρ = −0.5, as

out-of-the-money options with negative correlation is more relevant in real-life applications.

In each sensitivity analysis, one parameter is increased or decreased by 10% from its base

value. We look at the effect of the parameter change on the following quantities: effective

discounting e−rT , effective total variance Σ, deterministic variance budget exceeding time T0,

risk-neutral expected time to exercise TE, the timer call price C, as well as the delta ∆.

Table 3 provides much information and we will only make a few comments. First, a plain-

vanilla option is usually more expensive with a longer maturity. Therefore, if we think of a

timer option as the average of a string of plain-vanilla options with different fixed maturities

and an average exercise time TE , then generally speaking the larger TE is, the more expensive

is the timer option. The expected exercise time intuitively is larger for smaller κ (since we

have V0 < θ), smaller θ, smaller V0, and larger budget B. These are all confirmed in the

table. Second, except for the case of changing B, the effective variance budget Σ changes

slowly with the parameters, explaining why a first-order approximation in η for Σ gives very

accurate results. Third, the higher η is, the more optionality the timer call has, and the more

expensive the price is. Increasing volatility also increases the expected exercise time. Finally,

26

the behavior of the delta is similar to that of a plain-vanilla option. The delta gets higher with

higher call option price.

In unreported work, we also perform a sensitivity analysis for the 3/2 model. The results

are similar to those in the Heston model discussed above.

4.2 The Black-Scholes Implied Volatility Surface

The Black-Scholes implied volatility surface for plain-vanilla options provides a quick way to

appreciate the pricing behavior of a general stochastic volatility model. It also allows easier

comparison of different models. We can define such a surface for timer options as well. We

assume that interest rate r and dividend rate δ are not simultaneously zero. For any fixed

variance budget B and strike K, we compute the time Teff such that the timer call option can

be viewed as a plain-vanilla call option with a fixed maturity Teff . That is, Teff solves

C(S, ξ, V ) = Se−δTeffN(d+eff )−Ke−rTeffN(d−eff ), (145)

with

d±eff =log(S/K) + (r − δ)Teff√

B − ξ± 1

2

√B − ξ. (146)

Here we will not be concerned with the general existence or uniqueness properties of Teff . For

the zero dividend rate case we examine here, it is easy to show that Teff is always uniquely

defined.

Once we get Teff , we define the Black-Scholes implied volatility σimp(B − ξ,K) as

σimp(B − ξ,K) ≡√

B − ξ

Teff. (147)

The motivation is as follows. In the Black-Scholes framework, volatility is constant, so the

exercise time of the timer call option is a constant determined by the remaining variance

budget B − ξ. Therefore, σimp(B − ξ,K) under the Black-Scholes framework gives us the

same timer option price C(S, ξ, V ) in a stochastic volatility model. Such a quantity is very

useful for practical trading and quoting purposes because it makes price comparison across

different variance budgets, strikes and underlyings on more equal footing.

For simplicity, we will assume that currently ξ = 0. If it is not zero, we just modify B

to be the remaining variance budget B − ξ. The function σimp(B,K) gives us an implied

volatility surface for timer options. This implied volatility surface is different from the usual

implied volatility surface for plain-vanilla options due to the different nature of timer and plain-

vanilla options. In particular, we see that the implied volatility here enters into the pricing

27

equation (145) through the two discounting factors e−δTeff and e−rTeff , rather than through the

total variance, as is the case for plain-vanilla options.

The extremely fast and accurate approximation we have established allows us to plot such

implied volatility surface for timer options under stochastic volatility model with little cost in

computing time. In Figure 1, we plot the implied volatility surface for timer options under

the Heston model. The left and right subplots are for ρ = −0.5 and ρ = 0.5, respectively. The

other parameters used are the same as those in Table 1. Figure 2 plots the implied volatility

surface under the 3/2 model with parameters the same as those in Table 2. In both figures,

the range for strike is from 80 to 120, and the variance budget range goes from 0.05 to 0.5,

roughly corresponding to a target expected exercise time of 5.5 years.

The timer call prices in Tables 1 and 2 with the chosen parameters are very close for the

three values of K and three values of ρ reported. Figures 1 and 2 show that the implied

volatility surfaces from these two models also share some similarities. Both models exhibit

negative skew for negative ρ and positive skew for positive ρ. Also, the skew in both models

when plotted against strike is more prominent for small variance budget. However, these

two models also exhibit some remarkable differences when we consider the term structure of

implied volatility as a function of variance budget with fixed strike level. The Heston model

has a dip around B = 0.1 while the term structure in the 3/2 model has a more simple

monotonic structure. In the positive correlation case, the 3/2 model has a downward sloping

term structure for out-of-the-money timer calls, while in the Heston model, the term structure

is U -shaped and eventually increasing. As stochastic volatility models are often calibrated

using plain-vanilla option prices with maturities up to about two years, the above result shows

that there might be considerable model risk when valuing timer options with large variance

budget, especially when the parameter calibration process uses few maturities.

5 Conclusion

We have developed a perturbation technique for pricing timer options under general stochastic

volatility models. For the special cases of the Heston model and the 3/2 model, we obtain

very intuitive Black-Scholes-like closed-form formulas. Numerical analysis shows that these

formulas are very accurate and extremely fast. This offers a considerable advantage over

computationally expensive methods such as high-dimensional numerical integration or Monte

Carlo. Besides being fast and accurate, our method also has many other attractive features.

There are many research directions one can take following our approach. First, it is useful

to consider other stochastic volatility models. Because of the general solution technique in this

28

paper, we should be able to get analytic formula for the timer options if the drift and diffusion

functions of the variance process are sufficiently simple. Second, our method requires that the

volatility coefficient η of the variance process be small. This is an intrinsic shortcoming of the

perturbation method. It is therefore useful to study the behavior of the approximation for

relatively large η in order to improve the current approximation or to redesign a better ap-

proximation. Third, it is useful to design an approximation for timer options with a maximum

mandated expiry. This is an important feature for some timer options in practice. However, it

seems difficult to extend the current PDE perturbation technique to this case. We leave these

to future exploration.

29

6 Appendix

6.1 Verifying Equation (18)

For this purpose, let us denote the deterministic variance process at time t with initial value

V at initial time 0 by v(t, V ), where v(0, V ) = V . Notice that we have

dv(t, V )

dt= a(v(t, V )). (148)

Integrating, we get

t =

∫ v(t,V )

V

1

a(u)du. (149)

The above equation also verifies that v is a function of t and V . Differentiating with respect

to V , we get

∂v(t, V )

∂V=

a(v(t, V ))

a(V ). (150)

Differentiating the defining equation for T = T (ξ, V )

∫ T

0v(s, V ) ds = B − ξ (151)

with respect to V and ξ, we get by using equation (150) that

TV ≡ ∂T

∂V= − 1

v(T, V )

∫ T

0

∂

∂Vv(s, V ) ds = − 1

v(T, V )· v(T, V )− V

a(V ), (152)

Tξ ≡∂T

∂ξ= − 1

v(T, V ). (153)

It is now easy to verify that V Tξ + a(V )TV + 1 = 0.

6.2 Method of Characterisitcs for Solving Equation (57)

Pick any constant V ∗ > 0 and any smooth function Φ(·). Define

z0 =V − V ∗

V ∗, (154)

z = Φ

(B − ξ +

∫ V

V ∗

u

A(u)du

). (155)

Any constant V ∗ will work, but in practice stochastic volatility models are often mean-reverting

and it is often convenient to choose V ∗ to be the long-run mean so that we do not introduce

an unnecessary constant. We will always choose the function Φ such that when B = ξ, we

30

have z0 = z. This will usually pick up a unique form for Φ. Now switch to the new variables

by defining

Q(z, z0) = Q(ξ, V ). (156)

Because V zξ + A(V )zV = 0, it can be worked out that the PDE for Q(ξ, V ) now reduces to

an ODE for Q(z, z0):

Qz0 =V ∗

A(V ∗(1 + z0))q(z, z0), (157)

where q(z, z0) = q(ξ, V ). With the specific choice of Φ, the boundary condition becomes

Q(z0, z0) = 0. (158)

Therefore, the solution of Q is then simply

Q(ξ, V ) = Q(z(ξ, V ), z0(V )) =

∫ z0

z

V ∗

A(V ∗(1 + z0))q(z, z0) dz0. (159)

The only potential difficulty in the above method is that the function Φ might be given as an

implicit function. However, in the Heston model, Φ can be written explicitly using the product

log function, and in the 3/2 model, Φ is explicit.

6.3 Solving the PDEs in the Heston Model

6.3.1 Solving T0 in the Heston Model

By definition, T0(ξ, V ) is the first time the variance budget B is exceeded with initial variance

V and initial accumulated variance ξ. In the Heston model when η = 0, variance process v(t)

is deterministic

v(t) = V + (V − θ)e−κt, (160)

with initial variance v(0) = V . Therefore, the integrated variance process ξt with initial value ξ

is given by

ξt = ξ +

∫ t

0v(s) ds = ξ + θt+ (V − θ)

1− e−κt

κ. (161)

Letting ξt = B then gives us the solution in equation (65).

By the implicit function theorem, the derivatives of T0 are given by

T0,V = −1− e−κT0

κ

1

θ + (V − θ)e−κT0, (162)

T0,ξ = − 1

θ + (V − θ)e−κT0. (163)

31

It’s straightforward to verify that this implicit solution satisfies the PDE in equation (63).

The same implicit solution could also be obtained using Lagrange’s method of characteris-

tics for first-order PDEs. The characteristics are given by the solution of

dξ

V=

dV

κ(θ − V )=

dT0

−1. (164)

Solving these equations will give us two integral constants which when coupled with the bound-

ary condition gives us the same implicit solution in the main text. We omit the details here.

We now express T0 in terms of the product log function. From the implicit solution for T0

and the definition for z0, we have

κT0 + z0(1− e−κT0

)= κ

B − ξ

θ. (165)

Rearranging, we get

−κT0 + z0e−κT0 = z0 − κ

B − ξ

θ. (166)

Taking exponentials of both sides and multiplying by z0, we get

z0e−κT0ez0e

−κT0= z0e

z0e−κB−ξ

θ . (167)

If we define

z = z0e−κT0 , (168)

we get from equation (167) that

z = W(z0e

z0e−κB−ξ

θ

). (169)

The solution T0 is then given by

T0 =1

κlog

z0z. (170)

This solution has the shortcoming of a removable singularity at z0 = z = 0. From equation

(167), we have

z0z

= ez−z0eκB−ξ

θ . (171)

Taking log of both sides, we get the alternative expression for T0

T0 =z − z0

κ+

B − ξ

θ. (172)

We can easily verify that when ξ = B, we get z = z0 so T0 = 0. Notice also that since κ > 0,

θ > 0, and B ≥ ξ, z is always real. It can also be seen that z and z0 will always have the same

32

sign. In addition, we always have |z0| ≥ |z| since the function xex is an increasing function on

the real line. That is, we have R ≥ 1. When V > θ, we have z0 ≥ z > 0. When V < θ, we

have z0 ≤ z < 0. The above facts imply that T is always well-defined and nonnegative. The

only case for T0 = 0 is when ξ = B.

The quantity z has the meaning of percentage deviation of instantaneous variance at time T0

from the long-run mean θ, in particular, we always have 1 + z > 0. The variable z0 is

the percentage deviation of the current V from the long-run mean θ. In the Heston model,

regardless of whether η is 0 or not, this percentage deviation decays at the rate κ, which is

precisely what equation (67) is saying.

6.3.2 Solving T to Second Order in η in the Heston Model

We need to solve the first-order PDE

Hξ + κθ − V

VHV +

1

2

[T0,V V − r(T0,V )

2]= 0. (173)

First we need the expression for the second-order derivative T0,V V . Using the fact that

dW (x)

dx=

W (x)

x(1 +W (x)), (174)

we can express the derivatives of T0 in terms of z and z0 as

T0,ξ = −1

θ

1

1 + z, (175)

T0,V =z − z0

θκ(1 + z)z0, (176)

T0,ξξ =κz

(1 + z)3θ2, (177)

T0,ξV =z(1 + z0)

(1 + z)3z0θ2, (178)

T0,V V =z(2z − 2z0 + z2 − z20)

κ(1 + z)3z20θ2

. (179)

This suggests a change of coordinates from (ξ, V ) to (z, z0). Simple calculation shows that

∂z

∂ξ=

κz

θ(1 + z), (180)

∂z

∂V=

z(1 + z0)

θz0(1 + z), (181)

∂z0∂ξ

= 0, (182)

∂z0∂V

=1

θ. (183)

33

Now define

H(z, z0) = H(ξ, V ). (184)

By the chain rule,

Hξ =κz

θ(1 + z)Hz, (185)

HV =1

θHz0 +

z(1 + z0)

θz0(1 + z)Hz. (186)

Notice that

θ − V

V= − z0

1 + z0. (187)

Simple calculation then shows that

Hξ + κθ − V

VHV = − κz0

θ(1 + z0)Hz0 . (188)

The PDE for H then becomes an ODE for H:

Hz0 =θ(1 + z0)

κz0

T0,V V − r(T0,V )2

2. (189)

Using the boundary condition H(z0, z0) = 0, we get

H(z, z0) =

∫ z0

z

θ(1 + u)

κu

T0,V V (z, u) − r(T0,V (z, u))2

2du. (190)

It is now a simple matter of integration to get the expression in equation (81).

The above method of solving the first-order PDE is actually very general. For any source

term which can be written as V f(z, z0), the solution to

V Fξ + κ(θ − V )FV + V f(z, z0) = 0 (191)

with boundary condition F (B,V ) = 0 is given by

F =

∫ z0

z

θ(1 + u)

κuf(z, u) du. (192)

In particular, since V = θ(1 + z0), by taking f(z, z0) = 1/V , we can also verify that

T0(z, z0) =

∫ z0

z

θ(1 + u)

κu

1

θ(1 + u)du =

1

κlog

z0z

(193)

gives us the solution for T0.

34

6.3.3 Solving Σ2 to First Order in η in the Heston Model

The zeroth-order solution is simply

Σ2 = Σ20(ξ, V ) = B − ξ. (194)

For nonzero η, we postulate the following solution form

Σ2 = B − ξ + 2ηρ(r − δ)G(ξ, V ). (195)

Plugging this trial solution in, we get a Cauchy problem for G = G(ξ, V ):

V Gξ + κ(θ − V )GV + V T0,V = 0, (196)

with the boundary condition

G(B,V ) = 0. (197)

Notice that the technique in solving H above is valid for any source term. Therefore, if we

define G(z, z0) = G(ξ, V ), we can immediately write the solution as

G(z, z0) =

∫ z0

z

θ(1 + u)

κuT0,V (z, u) du (198)

=1

κ2(1 + z)

∫ z0

z

(1 + u)(z − u)

u2du. (199)

Integrating, we get

G(ξ, V ) =(1−R)(Rz − 1) +R(z − 1) logR

κ2R(1 + z). (200)

6.4 Solving the PDEs in the 3/2 Model

6.4.1 Solving T0 in the 3/2 Model

When η = 0, the variance process v(t, V ) with initial value V at time 0 is deterministic

v(t, V ) =V θ

(θ − V )e−κθt + V. (201)

A more intuitive way to understand the above equation is

1

v(t, V )=

(1

V− 1

θ

)e−κθt +

1

θ. (202)

This equation shows that when η = 0, the reciprocal of instantaneous variance decays expo-

nentially to its long-run level 1/θ with decaying rate κθ.

35

The deterministic time T0(ξ, V ) to hit the remaining variance budget B− ξ is given by the

solution of

∫ T

0v(t, V ) dt = B − ξ. (203)

The above gives the solution for T0 in the main text.

Below, we provide here a more general method by switching to two new variables z and z0.

This approach is useful for solving the higher-order PDEs for T , T ′ and Σ2.

Similar to the situation in the Heston model, we define

z0 ≡V − θ

θ, (204)

z ≡ V − θ

θe−κ(B−ξ), (205)

R ≡ z0z

= eκ(B−ξ). (206)

Notice that we always have R ≥ 1.

The following derivatives are needed when we switch from the variables (ξ, V ) to (z, z0):

∂z

∂ξ= κz, (207)

∂z

∂V=

z

θz0, (208)

∂z0∂ξ

= 0, (209)

∂z0∂V

=1

θ. (210)

For any function F (ξ, V ), define F (z, z0) = F (ξ, V ). It can be easily worked out that

V Fξ + κV (θ − V )FV = −κθz0(1 + z0)Fz0 . (211)

Therefore, for any function f(z, z0), the solution to

V Fξ + κV (θ − V )FV + f(z, z0) = 0 (212)

with boundary condition F (B,V ) = 0 is given by a simple integration

F (ξ, V ) = F (z, z0) =

∫ z0

z

1

κθu(1 + u)f(z, u) du. (213)

Taking f(z, z0) = 1 and integrating, we get

T0 =1

κθlog

(1 + z

z

z01 + z0

)=

1

κθlog

(R(1 + z)

1 +Rz

). (214)

36

Similarly, we have

T ′0 =

1

κ′θ′log

(1 + z′

z′z′0

1 + z′0

)=

1

κ′θ′log

(R′(1 + z′)

1 +R′z′

), (215)

with

z′0 ≡V − θ′

θ′, (216)

z′ ≡ V − θ′

θ′e−κ′(B−ξ), (217)

R′ ≡ z′0z′

= eκ′(B−ξ). (218)

6.4.2 Solving T to Second Order in η in the 3/2 Model

The derivatives we need here are

T0,V =z − z0

κθ2z0(1 + z0)(1 + z), (219)

T0,V V =(z0 − z)(z + z0 + 2zz0)

κθ3z20(1 + z0)2(1 + z)2. (220)

The quantity H(ξ, V ;κ, θ, r) satisfies the following PDE

V Hξ + κV (θ − V )HV +1

2V 3[T0,V V − r(T0,V )

2]= 0. (221)

Using V = θ(1 + z0), we can write the source term in terms of z and z0:

1

2V 3[T0,V V − r(T0,V )

2]= h(z, z0). (222)

The solution H is then given by the integration

H =

∫ z0

z

1

κθu(1 + u)h(z, u) du. (223)

Though tedious, the algebra is simple, and we omit the details here. In the main text, we

give H in terms of R = eκ(B−ξ) and V which is closer to our original variables ξ and V .

6.4.3 Solving Σ2 to First Order in η in the 3/2 Model

Write

Σ2 = B − ξ + 2ηρ(r − δ)G(ξ, V ). (224)

The PDE we need to solve for G = G(ξ, V ) is:

V Gξ + κV (θ − V )GV + V 2T0,V = 0. (225)

Writing V 2T0,V as a function of z and z0, this equation can be solved using the same technique

above, and we omit the details here.

37

References

D.H. Ahn and B. Gao. 1999. A parametric nonlinear model of term structure dynamics. Review

of Financial Studies 12, 721–762.

Y. Aıt-Sahalia. 1999. Transition densities for interest rate and other nonlinear diffusions. Jour-

nal of Finance 54, 1361–1395.

Y. Aıt-Sahalia. 2002. Maximum-likelihood estimation of discretely-sampled diffusions: A

closed-form approximation approach. Econometrica 70, 223–262.

Y. Aıt-Sahalia. 2008. Closed-form likelihood expansions for multivariate diffusions. Annals of

Statistics 36, 906–937.

L.B.G. Andersen, and V.V. Piterbarg. 2007. Moment explosions in stochastic volatility models.

Finance and Stochastic 11, 29-50.

C. Bernard, Z. Cui. 2011. Pricing timer options. Journal of Computational Finance 15(1),

69–104.

A. Bick. 1995. Quadratic-variation-based dynamic strategies. Management Science 41(4), 722–

732.

P. Carr, and R. Lee. 2010. Hedging variance option on continuous semimartingales. Finance

and Stochastic 14, 179–207.

D. Hawkins, S. Krol. 2008. Product overview: Timer options. Lehman brothers Equity Deriva-

tives note.

S.L. Heston. 1993. A closed-form solution for options with stochastic volatility with applica-

tions to bond and currency options. Review of Financial Studies 6, 327–343.

S. Karlin, and H.M. Taylor. 1981. A Second Course in Stochastic Processes. Academic Press,

New York.

E. Kirk. 1995. Correlations in the energy markets, in managing energy price risk. Risk Publi-

cations and Enron.

R. Lee. 2012. Timer options for risk-controlled variance exposure. Global Derivatives USA

2012 conference presentation.

A.L. Lewis. 2000. Option Valuation Under Stochastic Volatility. Finance Press, Newport Beach,

California.

C.X. Li. 2010. Managing volatility risk. Doctoral Dissertation, Columbia University.

M. Li. 2010. A damped diffusion framework for financial modeling and closed-form maximum

likelihood estimation. Journal of Economic Dynamics and Control 34, 132–157.

38

M. Li, S. Deng, and J. Zhou. 2008. Closed-form approximations for spread option prices and

greeks. Journal of Derivatives 15(3), 58–80.

M. Li, S. Deng, and J. Zhou. 2010. Multi-asset spread option pricing and hedging. Quantitative

Finance 10(3), 305–324.

L.Z.J. Liang, D. Lemmens, and J. Tempere. 2011. Path integral approach to the pricing of

timer options with the Duru-Kleinert time transformation. Physical Review E (83).

A. Lipton. 2001. Mathematical Methods for Foreign Exchange: a Financial Engineers Ap-

proach. World Scientific.

A. Neuberger. 1990. Volatility trading. Working paper. London Business School.

D. Saunders. 2010. Pricing timer options under fast mean-reverting stochastic volatility. Work-

ing paper.

N. Sawyer. 2007. SG CIB launches timer options. Risk 20(7), 6.

N. Sawyer. 2008. Equity derivative house of the year: Societe Generale. Risk 21(1).

Zeliade Systems. 2011. Heston 2010. Zeliade Systems whitepaper.

39

Table 1: Accuracy of Timer Call Price Approximation in the Heston Model

Parameters used here are the same ones as in Liang, Lemmens and Tempere (2011): V0 = 0.087,κ = 2, θ = 0.09, η = 0.375, B = 0.087, ξ = 0, r = 0.015, S0 = 100, and δ = 0. Analytical resultsare taken from Liang, Lemmens and Tempere (2011). In Panel A, we give the timer option pricesassuming that r = 0 and η = 0, where we have exact pricing formulas. In Panel B, we compare theanalytic results with the approximation in this paper. Approx1 is price obtained if we solve T , T ′

and Σ all to first order in η. In Approx1-2, Σ is solved exactly to first order in η, while T and T ′

are solved exactly to second order in η. All errors reported are percentage errors with respect tothe analytic results.

Panel A: Naive Approximations with r = 0 or η = 0

K ρ Analytic r = 0 Error η = 0 Error

90 −0.5 17.8095 16.8371 −5.46% 17.6148 −1.09%0 17.7249 16.8371 −5.01% 17.6148 −0.62%

0.5 17.6263 16.8371 −4.48% 17.6148 −0.07%

100 −0.5 12.5789 11.7263 −6.78% 12.3837 −1.55%0 12.4772 11.7263 −6.02% 12.3837 −0.75%

0.5 12.3691 11.7263 −5.20% 12.3837 0.12%

110 −0.5 8.6515 7.9445 −8.17% 8.4697 −2.10%0 8.5449 7.9445 −7.03% 8.4697 −0.88%

0.5 8.4393 7.9445 −5.86% 8.4697 0.36%

Panel B: Singular Perturbation Method

K ρ Analytic Approx1 Error Approx1-2 Error

90 −0.5 17.8095 17.7033 −0.60% 17.8167 0.04%0 17.7249 17.6148 −0.62% 17.7287 0.02%

0.5 17.6263 17.5258 −0.57% 17.6400 0.08%

100 −0.5 12.5789 12.4847 −0.75% 12.5815 0.02%0 12.4772 12.3837 −0.75% 12.4806 0.03%

0.5 12.3691 12.2817 −0.71% 12.3788 0.08%

110 −0.5 8.6515 8.5719 −0.92% 8.6500 −0.02%0 8.5449 8.4697 −0.88% 8.5476 0.03%

0.5 8.4393 8.3666 −0.86% 8.4444 0.06%

40

Table 2: Accuracy of Timer Call Price Approximation in the 3/2 Model

Parameters used here are the same ones as in Liang, Lemmens and Tempere (2011): V0 = 0.2952,κ = 22.84, θ = 0.46692, η = 8.56, B = V0, ξ = 0, r = 0.015, S0 = 100, and δ = 0. Analytical resultsare taken from Liang, Lemmens and Tempere (2011). In Panel A, we give the timer option pricesassuming that r = 0 and η = 0, where we have exact pricing formulas. In Panel B, we compare theanalytic results with the approximation in this paper. Approx1 is price obtained if we solve T , T ′

and Σ all to first order in η. In Approx1-2, Σ is solved exactly to first order in η, while T and T ′

are solved exactly to second order in η. All errors reported are percentage errors with respect tothe analytic results.

Panel A: Naive Approximations with r = 0 or η = 0

K ρ Analytic r = 0 Error η = 0 Error

90 −0.5 17.8064 16.8371 −5.44% 17.2853 −2.93%0 17.7046 16.8371 −4.90% 17.2853 −2.37%

0.5 17.5839 16.8371 −4.25% 17.2853 −1.70%

100 −0.5 12.5780 11.7263 −6.77% 12.1045 −3.76%0 12.4619 11.7263 −5.90% 12.1045 −2.87%

0.5 12.3300 11.7263 −4.90% 12.1045 −1.83%

110 −0.5 8.6518 7.9445 −8.18% 8.2461 −4.69%0 8.5339 7.9445 −6.91% 8.2461 −3.37%

0.5 8.4026 7.9445 −5.45% 8.2461 −1.86%

Panel B: Singular Perturbation Method

K ρ Analytic Approx1 Error Approx1-2 Error

90 −0.5 17.8064 17.3662 −2.47% 17.7653 −0.23%0 17.7046 17.2853 −2.37% 17.6856 −0.11%

0.5 17.5839 17.2038 −2.16% 17.6053 0.12%

100 −0.5 12.5780 12.1962 −3.04% 12.5356 −0.34%0 12.4619 12.1045 −2.87% 12.4443 −0.14%

0.5 12.3300 12.0120 −2.58% 12.3522 0.18%

110 −0.5 8.6518 8.3383 −3.62% 8.6113 −0.47%0 8.5339 8.2461 −3.37% 8.5188 −0.18%

0.5 8.4026 8.1532 −2.97% 8.4255 0.27%

41

Table 3: Sensitivity Analysis of Timer Call Prices in Heston Model

Base case parameters are the same ones as in Liang, Lemmens and Tempere (2011): V0 = 0.087,κ = 2, θ = 0.09, η = 0.375, B = 0.087, ξ = 0, r = 0.015, S0 = 100, and q = 0. Here for thebase case, we use K = 110 and ρ = −0.5. In each sensitivity analysis, one parameter is increasedor decreased by 10% from its base value. We look at the effect of the parameter change on thefollowing quantities: effective discounting e−rT , effective total variance Σ, deterministic variancebudget exceeding time T0, risk-neutral expected time to exercise TE, the timer call price C, as wellas the delta ∆.

parameter e−rT Σ T0 TE C ∆

base case 0.9833 0.0885 0.9810 1.1228 8.6500 0.4542

κ +10% 0.9836 0.0885 0.9801 1.1041 8.6349 0.4538−10% 0.9830 0.0886 0.9820 1.1451 8.6674 0.4548

θ +10% 0.9844 0.0884 0.9300 1.0514 8.5992 0.4527−10% 0.9820 0.0887 1.0416 1.2100 8.7125 0.4562

η +10% 0.9829 0.0887 0.9810 1.1526 8.6766 0.4550−10% 0.9837 0.0884 0.9810 1.0959 8.6249 0.4536

ρ +10% 0.9833 0.0887 0.9810 1.1228 8.6601 0.4544−10% 0.9833 0.0884 0.9810 1.1228 8.6398 0.4541

V0 +10% 0.9839 0.0883 0.9398 1.0815 8.6144 0.4532−10% 0.9827 0.0884 1.0233 1.1648 8.6653 0.4550

B +10% 0.9817 0.0973 1.0781 1.2281 9.2786 0.4639−10% 0.9849 0.0795 0.8838 1.0160 7.9729 0.4431

r +10% 0.9816 0.0885 0.9810 1.1227 8.7109 0.4565−10% 0.9850 0.0882 0.9810 1.1229 8.5688 0.4517

42

00.1

0.20.3

0.40.5

80

90

100

110

120

0.285

0.29

0.295

0.3

0.305

0.31

0.315

0.32

BK0

0.10.2

0.30.4

0.5

80

90

100

110

1200.25

0.255

0.26

0.265

0.27

0.275

0.28

0.285

0.29

BK

Figure 1: Black-Scholes implied volatility surface for timer options under the Heston

model. The x and y axes are the variance budget B and strike level K. The left and right subplotsare for ρ = −0.5 and ρ = 0.5, separately. The other parameters used are the same as those inTable 1. The precise definition of Black-Scholes implied volatility for timer options is given inequation (147).

43

00.1

0.20.3

0.40.5

80

90

100

110

120

0.29

0.3

0.31

0.32

0.33

BK0

0.10.2

0.30.4

0.5

80

90

100

110

1200.245

0.25

0.255

0.26

0.265

0.27

0.275

0.28

0.285

BK

Figure 2: Black-Scholes implied volatility surface for timer options under the 3/2 model.

The x and y axes are the variance budget B and strike level K. The left and right subplots are forρ = −0.5 and ρ = 0.5, separately. The other parameters used are the same as those in Table 2. Theprecise definition of Black-Scholes implied volatility for timer options is given in equation (147).

44